The angle of the top tube has no bearing on the frame geometry, even though it's called "compact geometry". You only contact the bike via the pedals, seat, and bars. How those points are connected doesn't matter for bike fit, only style preference (and standover height, which is not an issuse for most road riders).
"relaxed geometry" means a taller head tube (allowing the bars to go higher) and sometimes a shorter top tube (allowing the bars to be closer to the rider). What you need will depend on your body shape and how low and stretched out you want to be. A good fitter can work with you to figure out the proper position for you.
I'd also look into strengthening your back and core muscles so they supprort your spine better. A road bike is only faster than a hybrid because the position is more aerodynamic, i.e. the bars are lower. If you want that speed you're going to have to figure out how make it comfortable.